A characterization of a class of dynamical systems in a collection of L ( x 1 < / sub > x 2 sub > ) were made to promote the study nature and obtained degrees continuous system . Also use the concept of scope and category theory functor in depth discussion on the relationship between the power system and its enveloping semigroup . In the first chapter introduces the necessary definitions and notations . Described in [ 4 ] and [ 5 ] In the second chapter , a collection of classes L (x 1 , x 2 ) was further discussed . First , its definition made ??the promotion , and research related to the nature of the collection class promotion , and then given a characterization of the equicontinuous system . Second, the collection L ( x 1 < / sub > , x 2 < / sub > ) , L (x 1 sub > , x 2 < / sub > , x 3 ), L (x 1 , x 2 , x 3 , x 4 sub >) , ... , L ( x 1 < / sub > , ... , X n < / sub > ) , the relationship between the discussion, and to give some examples . In the third chapter , the concept of the use of category theory in the category and functor , define the areas of power system the envelope the semigroup areas E the covariant functor F 1 as well as areas T to T to the scope of E * anti- change functor F 2 sub > . In addition , and also discuss the package of the product of the system in the areas of T enveloping semigroup and category E enveloping semigroup direct product consistency and the areas T in inverse limit system the enveloping semigroup areas E network semigroup inverse limit consistency .